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\newtheorem{theorem}{Theorem}
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\begin{center}
\vskip 1cm{\LARGE\bf Cipolla Pseudoprimes}
\vskip 1cm
\large
Y. Hamahata\footnote{Partially supported by Grant-in-Aid for
Scientific Research (No.\ 18540050),
Japan Society for the Promotion of Science.}
and Y. Kokubun\\
Department of Mathematics \\
Tokyo University of Science \\
Noda, Chiba, 278-8510 \\
Japan \\
{\tt hamahata\_yoshinori@ma.noda.tus.ac.jp}\\
\end{center}
\vskip .2in
\begin{abstract}
We consider the pseudoprimes that M. Cipolla
constructed. We call such pseudoprimes {\it Cipolla pseudoprimes}.
In this paper we find infinitely many
Lucas and Lehmer pseudoprimes that are analogous to
Cipolla pseudoprimes.
\end{abstract}
\section{Introduction}
Take an integer $a>1$.
A {\it pseudoprime to base} $a$ is a composite number
$n$ such that
$a^{n-1}\equiv 1\, (\bmod\, n)$.
In 1904, M. Cipolla \cite{Cipolla} found infinitely many pseudoprimes
to a given base $a$. To be more precise,
\begin{theorem}[Cipolla \cite{Cipolla}, cf.\ Ribenboim \cite{Ribenboim1}]\label{1}
Let $p$ be a prime such that
$p$ does not divide $a(a^2-1)$. Put
$$
n_1=\frac{a^p-1}{a-1},\quad
n_2=\frac{a^p+1}{a+1},\quad
n=n_1n_2.
$$
Then $n$ is a pseudoprime to base $a$.
\end{theorem}
In this paper we call such $n$ a
{\it Cipolla pseudoprime}. In the above theorem,
if we set $P=a+1$, $Q=a$, then
$n$ is written as $n=U_{2p}/P$,
where $U_{2p}$ is a term in the Lucas sequence
with parameters $P$ and $Q$. See the next
section for Lucas sequences.
From this observation, the following question
arises.
For given integers $P, Q$,
are there infinitely many
Lucas pseudoprimes with parameters $P$ and $Q$
of the form $U_{2p}/P$?
Here Lucas pseudoprimes will be defined in the
next section.
\par
The purpose of the paper is to
solve the above question affirmatively
under a certain condition.
As a corollary to our result, we derive the
result of Lehmer \cite{Lehmer2}.
We are also going to consider an
analogous question for Lehmer sequences.
\section{Cipolla-Lucas pseudoprimes}
In this section we consider Lucas pseudoprimes
of special type.
\par
Let $P, Q$ be integers such that
$D:=P^2-4Q\ne 0$, and
$\alpha , \beta$ the roots of
the polynomial
$z^2-Pz+Q$.
For a nonnegative integer $n$, put
$$
U_n=\frac{\alpha^n-\beta^n}{\alpha -\beta},
\quad V_n=\alpha^n+\beta^n.
$$
For example, we have
$U_0=1$, $U_1=1$, $U_2=P$,
$V_0=2$, and $V_1=P$.
One sees that
$(U_n)_{n\geq 0}$ and $(V_n)_{n\geq 0}$
are integer sequences.
We call the sequences
$(U_n)_{n\geq 0}$, $(V_n)_{n\geq 0}$
the {\it Lucas sequences with parameters $P$ and $Q$}.
\par
We exhibit some results needed afterwards.
One can consult Ribenboim
\cite{Ribenboim1,Ribenboim2}
for the basic results.
\par
\noindent
(I) For a nonnegative integer $n$,
$U_{2n}=U_nV_n$.
\par
\noindent
(II) (a) If $P$ is odd and $Q$ is even,
then $U_n$, $V_n$ ($n\geq 1$)
are odd.
\par
(b) If $P$ and $Q$ are odd, then
$U_n$, $V_n$ ($3\nmid n$) are odd.
\par
\noindent
(III) (a) When $U_m\ne 1$,
$U_m|U_n$ if and only if $m|n$.
\par
(b) When $V_m\ne 1$,
$V_m|V_n$ if and only if $m|n$ and $n/m$ is odd.
\par
\noindent
(IV) For any odd prime $p$,
\begin{eqnarray}
2^{p-1}U_p&=&\sum_{k=0}^{(p-1)/2}
\left(\begin{array}{c}p\\ 2k+1\end{array}\right)
P^{p-(2k+1))}D^k, \\
2^{p-1}V_p&=&\sum_{k=0}^{(p-1)/2}
\left(\begin{array}{c}p\\ 2k\end{array}\right)
P^{(p-2k)}D^k.
\end{eqnarray}
$$
U_p\equiv \left(\frac{D}{p}\right) \pmod p,
\quad
V_p\equiv P \pmod p .
$$
\par
We recall Lucas pseudoprimes.
A composite number $n$ is a
{\it Lucas pseudoprime with parameters $P$ and $Q$} if
$$
U_{n-\left(\frac{D}{n}\right)}
\equiv 0 \pmod n
$$
holds. Here $\left(\frac{D}{n}\right)$
is the Jacobi symbol.
\par
Now let us define an analogue of Cipolla
pseudoprimes for Lucas sequences.
\begin{definition}\label{2}{\rm
A composite number $n$ is called a
{\it Cipolla-Lucas pseudoprime with parameters}
$P$ and $Q$ if it is a Lucas pseudoprime with
parameters $P$ and $Q$ and has the form
$U_{2p}/P$ for a certain prime number $p$.
}
\end{definition}
\par
Our first result is as follows.
\begin{theorem}\label{3} Let $P$ be an odd number, and $Q$ a nonzero
integer such that ${\rm gcd}(P,Q)=1$.
Assume that $D=P^2-4Q$ is square-free.
Then there are infinitely many Cipolla-Lucas
pseudoprimes with parameters $P$ and $Q$.
\end{theorem}
\begin{proof}
Let $p$ be an odd prime such that
$\text{gcd}(p,3PD)=1$ and
$\varphi (D)|p-1$.
Then we show that $U_{2p}/P$ is a Lucas
pseudoprime. From now on we prove the theorem
step by step. Put $m=U_{2p}/P$.
\par
First of all, we prove
$m | U_{m-\left(\frac{D}{m}\right)}$.
Since $p$ is odd,
$U_p\equiv \left(\frac{D}{p}\right)\, (\bmod\, p),
\, V_p\equiv P\, (\bmod\, p)$.
So that
$$
U_{2p}=U_pV_p\equiv P\left(\frac{D}{p}\right)
\pmod p .
$$
Since $P=U_2$, $U_2|U_{2p}$, and
$\text{gcd}(p,P)=1$, we have
$m\equiv \left(\frac{D}{p}\right)
\, (\bmod\, p)$.
We recall that $P$ is odd and
$\text{gcd}(p,3)=1$.
Hence $U_p$ and $V_p$ are odd.
We see $2p| m-\left(\frac{D}{p}\right)$.
From this, we have
$U_{2p}| U_{m-\left(\frac{D}{p}\right)}$.
Moreover we have
$m|U_{m-\left(\frac{D}{p}\right)}$.
We prove
$\left(\frac{D}{p}\right) =
\left(\frac{D}{m}\right)$.
By (1) and (2),
\begin{eqnarray*}
2^{p-1}U_p
&\equiv &
pP^{p-1}\pmod D , \\
2^{p-1}V_p
&\equiv &
P^p\pmod D .
\end{eqnarray*}
By $\varphi (D)|p-1$, we have
$U_p\equiv p\, (\bmod\, D),\,
V_p\equiv P\, (\bmod\, D)$.
Hence $U_{2p}\equiv pP\, (\bmod\, D)$.
By $\text{gcd}(P,D)=1$,
it follows that
$m=U_{2p}/P\equiv p\,
(\bmod\, D)$.
Observe that $D\equiv 1\, (\bmod\, 4)$
because $P$ is odd.
Thus we have
$\left(\frac{D}{p}\right) =
\left(\frac{D}{m}\right)$, which implies
$m|U_{m-\left(\frac{D}{m}\right)}$.
\par
We next show that $m$ is a composite number.
Since $p$ is odd and $P=U_2=V_1$,
one has
$P\nmid U_p$ and $P|V_p$.
Now assume that
there exists an odd prime $p$
satisfying
$V_p=\pm P$.
Then one has $V_1|V_p$ and $V_p|V_1$.
This implies $p=1$, which is absurd.
Therefore $m$ is a composite number.
\par
Finally, we prove the infinitude of $U_{2p}/P$.
By Dirichlet's theorem on primes in arithmetic
progression, there are infinitely many primes $p$
such that $\varphi (D)|p-1$.
The number of primes $p$ with $\text{gcd}(p,3PD)> 1$
among them
is finite. This proves the claim.
\end{proof}
\vspace{2mm}
As a corollary to the last theorem, we can derive
a known result.
We call the Lucas sequence with parameters
$1$ and $-1$ the
{\it Fibonacci sequence}.
We write
$(F_n)_{n\geq 0}$
for it.
A composite number $n$ is called a
{\it Fibonacci pseudoprime} if
$$
F_{n-\left(\frac{D}{n}\right)}\equiv 0\pmod n
$$
is valid.
Using the last theorem, we have
\begin{corollary}[Lehmer \cite{Lehmer2}]\label{4}
There are infinitely many primes $p$ such that
$F_{2p}$ is a Fibonacci pseudoprime.
\end{corollary}
\begin{proof}
Since $P=1$ and $Q=-1$, $U_{2p}/P$ becomes $F_{2p}$.
In this case one has $D=5$.
Hence for any prime $p>5$ with
$p\equiv 1\, (\bmod\, 4)$, the two
conditions
$\text{gcd}(p, 3PD)=1$ and
$\varphi (D)|p-1$ hold.
This yields the result.
\end{proof}
\section{Cipolla-Lehmer pseudoprimes}
In this section we consider Lehmer pseudoprimes.
First, we review Lehmer sequences.
\par
Let $\alpha , \beta$ be distinct roots
of the polynomial
$f(z)=z^2-\sqrt{L}z+M$, where
$L>0$ and $M$ are rational integers,
and $K:=L-4M$ is the discriminant of
$f(z)$.
For a nonnegative integer $n$, put
\begin{eqnarray*}
D_n&=&\left\{
\begin{array}{cc}
(\alpha^n-\beta^n)/(\alpha -\beta) &
\text{if\, $n$\, is\, odd} \\
(\alpha^n-\beta^n)/(\alpha^2 -\beta^2) &
\text{if\, $n$\, is\, even},
\end{array}
\right. \\
E_n&=&\left\{
\begin{array}{cc}
(\alpha^n+\beta^n)/(\alpha +\beta) &
\text{if\, $n$\, is\, odd} \\
\alpha^n+\beta^n &
\text{if\, $n$\, is\, even}.
\end{array}
\right.
\end{eqnarray*}
For example, we have $D_0=0$, $D_1=D_2=1$,
$E_0=2$, $E_1=1$, and $E_2=L-2M$.
One sees that $(D_n)_{n\geq 0}$ and
$(E_n)_{n\geq 0}$ are integer sequences.
We call the sequences
$(D_n)_{n\geq 0}$ and
$(E_n)_{n\geq 0}$
the {\it the Lehmer sequences
with parameters $L$ and $M$}.
It should be noticed that
we modify the original definition of the Lehmer
sequences in order to
make them integer sequences.
\par
We exhibit some results needed afterwards.
One can consult Lehmer
\cite{Lehmer1} for the basic results.
\par
\noindent
(I) For a prime $p$,
$D_{2p}=D_pE_p$.
\par
\noindent
(II) $D_n$ is even in the following cases only
\par
(a) $L=4k$, $M=2l+1$, $n=2h$,
\par
(b) $L=4k+2$, $M=2l+1$, $n=4h$,
\par
(c) $L=4k\pm 1$, $M=2l+1$, $n=3h$.
\par
\noindent
(III) $E_n$ is even in the following cases only
\par
(a) $L=4k$, $M=2l+1$,
\par
(b) $L=4k+2$, $M=2l+1$, $n=2h$,
\par
(c) $L=4k\pm 1$, $M=2l+1$, $n=3h$.
\par
\noindent
(IV) If $m|n$, then $D_m|D_n$.
\par
\noindent
(V) For any odd prime $p$,
\begin{eqnarray}
2^{p-1}D_p&=&\sum_{k=0}^{(p-1)/2}
\left(\begin{array}{c}p\\ 2k+1\end{array}\right)
L^{(p-2k-1)/2}K^k, \\
2^{p-1}E_p&=&\sum_{k=0}^{(p-1)/2}
\left(\begin{array}{c}p\\ 2k\end{array}\right)
L^{(p-2k)/2}K^k.
\end{eqnarray}
$$
D_p\equiv \left(\frac{K}{p}\right)\pmod p ,
\quad
E_p\equiv \left(\frac{L}{p}\right)\pmod p .
$$
\par
Next, we review Lehmer pseudoprimes.
A composite number $n$ is called a
{\it Lehmer pseudoprime with parameters $L$ and $M$}
if
$$
D_{n-\left(\frac{KL}{n}\right)}
\equiv 0\pmod n
$$
holds. Here $\left(\frac{KL}{n}\right)$
denotes the Jacobi symbol.
\par
Any Cipolla pseudoprime is
written as $D_{2p}$ for some prime $p$.
Hence we define Lehmer pseudoprimes
related to Cipolla pseudoprimes
as follows.
\begin{definition}\label{5}{\rm
A composite number $n$ is called
a {\it Cipolla-Lehmer pseudoprime with
parameters $L$ and $M$}
if it is a Lehmer pseudoprime with
parameters $L$ and $M$ and has
the form $D_{2p}$
for a certain prime $p$}
\end{definition}
\par
Our second result is as follows.
\begin{theorem}\label{6}
Let $L$ be a square-free odd number and $M$ an integer
such that ${\rm gcd}(L,M)=1$. Assume that $K=L-4M$
is square-free.
Then there are infinitely many
Cipolla-Lehmer pseudoprimes with
parameters $L$ and $M$.
\end{theorem}
\begin{proof}
The proof is similar to that of
Theorem~\ref{3}.
Let $p$ be an odd prime such that
$\text{gcd}(p, KL)=1$
and $\varphi (KL)| p-1$.
Then we prove that $D_{2p}$
is a Lehmer pseudoprime.
Put $m=D_{2p}$.
\par
We first show $m| D_{m-\left(\frac{KL}{m}\right)}$.
Since $p$ is odd,
$D_{p}\equiv\left(\frac{K}{p}\right)\,
(\bmod\, p)$,
$E_{p}\equiv\left(\frac{L}{p}\right)\,
(\bmod\, p)$. Hence
$$
m=D_{2p}=D_pE_p\equiv
\left(\frac{KL}{p}\right)\pmod p .
$$
That is to say, $p|m-\left(\frac{KL}{p}\right)$.
Since $L$ is odd, $D_p$ and $E_p$ are odd.
Hence $m$ is odd.
We find that $m-\left(\frac{KL}{p}\right)$
is even.
Thus $2p|m-\left(\frac{KL}{p}\right)$.
Using this, we have $D_{2p}|D_{m-\left(\frac{KL}{p}\right)}$,
which shows $m|D_{m-\left(\frac{KL}{p}\right)}$.
We must prove $\left(\frac{KL}{p}\right) =
\left(\frac{KL}{m}\right)$.
Since
$K$ is odd, by (3) and (4),
\begin{eqnarray*}
2^{p-1}D_p&\equiv &
pL^{\frac{p-1}{2}}+K^{\frac{p-1}{2}}
\pmod {KL} ,\\
2^{p-1}E_p
&\equiv &
L^{\frac{p-1}{2}}+pK^{\frac{p-1}{2}}
\pmod {KL} .
\end{eqnarray*}
Since $\varphi (KL)|p-1$ and $2\nmid KL$
hold,
$2^{p-1}\equiv 1\, (\bmod\, KL)$.
Hence we have
$$
m=D_pE_p\equiv p\left(K^{p-1}
+L^{p-1}\right)\pmod {KL} .
$$
It should be noted that
$K^{p-1}+L^{p-1}
\equiv 1\,
(\bmod\, KL)$.
Indeed, because of $\text{gcd}(K,L)=1$,
the condition
$\varphi (K)\varphi (L)|p-1$
implies
$L^{p-1}\equiv 1\, (\bmod\, K)$
and
$K^{p-1}\equiv 1\, (\bmod\, L)$.
For any prime divisor $l$ of $K$,
$l|K^{p-1}+L^{p-1}-1$.
Hence we have
$K^{p-1}+L^{p-1}-1\equiv 0\,
(\bmod\, K)$.
In the same way, we have
$K^{p-1}+L^{p-1}-1\equiv 0\,
(\bmod\, L)$.
Therefore our claim is proven.
Using this observation, we obtain
$m\equiv p\, (\bmod\, KL)$.
By the way, we see
$KL=L^2-4ML\equiv L^2\equiv 1\, (\bmod\, 4)$.
Thus we conclude that $\left(\frac{KL}{p}\right)
=\left(\frac{KL}{m}\right)$.
We get
$m|D_{m-\left(\frac{KL}{m}\right)}$.
\par
Clearly $m=D_pE_p$ is a composite number.
\par
Finally we show the infinitude of $D_{2p}$.
By Dirichlet's theorem on primes in arithmetic
progression, there are infinitely many primes $p$
such that $\varphi (KL)|p-1$.
The number of primes $p$ with $\text{gcd}(p,KL)> 1$
among them
is finite. This proves the claim.
\end{proof}
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{\bf 31} (1930), 419--448.
\bibitem{Lehmer2}E. Lehmer,
\newblock On the infinitude of Fibonacci
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\newblock {\em Fibonacci Quart.}
{\bf 2} (1964), 229--230.
\bibitem{Ribenboim1}P. Ribenboim,
\newblock {\em The Book of Prime Number Records.}
\newblock Springer-Verlag, 1989.
\bibitem{Ribenboim2}P. Ribenboim,
\newblock {\em My Numbers, My Friends
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\newblock Springer-Verlag, 2003.
\bibitem{Rotkiewicz1}A. Rotkiewicz,
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\bibitem{Rotkiewicz2}A. Rotkiewicz,
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\end{thebibliography}
\bigskip
\hrule
\bigskip
\noindent
2000 {\it Mathematics Subject Classification}:
Primary 11A51; Secondary 11B39.\\
\noindent
{\it Keywords:} pseudoprime, Lucas sequence,
Lucas pseudoprime, Lehmer sequence,
Lehmer pseudoprime.
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received July 17 2007;
revised version received August 1 2007.
Published in {\it Journal of Integer Sequences}, August 14 2007.
\bigskip
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